text: Potential flow in two dimensions is simple to analyze using conformal mapping, by the use of transformations of the complex plane. However, use of complex numbers is not required, as for example in the classical analysis of fluid flow past a cylinder.  It is not possible to solve a potential flow using complex numbers in three dimensions.
  The basic idea is to use a holomorphic (also called analytic) or meromorphic function , which maps the physical domain  to the transformed domain . While , ,  and  are all real valued, it is convenient to define the complex quantities
  Now, if we write the mapping  as
  Then, because  is a holomorphic or meromorphic function, it has to satisfy the Cauchy–Riemann equations
  The velocity components , in the  directions respectively, can be obtained directly from  by differentiating with respect to . That is
  So the velocity field  is specified by
  Both  and  then satisfy Laplace's equation:
  So  can be identified as the velocity potential and  is called the stream function. Lines of constant  are known as streamlines and lines of constant  are known as equipotential lines (see equipotential surface).
  Streamlines and equipotential lines are orthogonal to each other, since
  Thus the flow occurs along the lines of constant  and at right angles to the lines of constant .
   is also satisfied, this relation being equivalent to . So the flow is irrotational. The automatic condition  then gives the incompressibility constraint .